A problem in mathematics is given to three students whose chances of solving it are 1/3, 1/4, and 1/5.

(i) What is the probability that the problem is solved?

(ii) What is the probability that exactly one of them will solve it?

To solve this problem, we can use the principle of probability. Let's break it down into two parts:

(i) What is the probability that the problem is solved?

To find the probability that the problem is solved, we need to calculate the complementary probability that none of the three students solves it and subtract it from 1.

The probability that the first student solves it is 1/3.
The probability that the second student solves it is 1/4.
The probability that the third student solves it is 1/5.

So, the probability that none of them solves it is:

(1 - 1/3) * (1 - 1/4) * (1 - 1/5) = (2/3) * (3/4) * (4/5) = 24/60 = 2/5

Now, to find the probability that the problem is solved, we subtract this from 1:

Probability(problem is solved) = 1 - Probability(problem is not solved) = 1 - 2/5 = 3/5

So, the probability that the problem is solved is 3/5.

(ii) What is the probability that exactly one of them will solve it?

To find the probability that exactly one of them will solve it, we can consider three cases:

Case 1: The first student solves it, and the other two do not.
The probability of this happening is (1/3) * (1 - 1/4) * (1 - 1/5) = (1/3) * (3/4) * (4/5) = 1/5.

Case 2: The second student solves it, and the other two do not.
The probability of this happening is (1 - 1/3) * (1/4) * (1 - 1/5) = (2/3) * (1/4) * (4/5) = 2/15.

Case 3: The third student solves it, and the other two do not.
The probability of this happening is (1 - 1/3) * (1 - 1/4) * (1/5) = (2/3) * (3/4) * (1/5) = 1/10.

Now, to find the total probability that exactly one of them will solve it, we sum up these probabilities:

Total probability(exactly one of them solves it) = Case 1 + Case 2 + Case 3 = 1/5 + 2/15 + 1/10 = 6/30 + 4/30 + 3/30 = 13/30

So, the probability that exactly one of them will solve it is 13/30.